We shall derive the necessary condition for the optimization problem in Section . This is done by investigating the sensitivity of the objective functional \(\mathcal {G}\)  with respect to shape variations. In particular, the shape variations are induced by the velocity method [1]}, and the shape sensitivity is analyzed using the rearrangement method which was formalized by Ito, K., Kunisch, K., and Peichl, G. in [2]}. In Section , we provide numerical algorithms through the help of the necessary conditions formulated in Section , and provide some illustrations of how these algorithms are implemented. To objectively observe the effects of the final shapes to fluid vortices, we look at the generation of twin-vortices in a dynamic nonlinear fluid flow. Finally, we draw conclusions and possible problems on the last section.
